Transient and Enduring Electronic Resonances Drive Coherent Long

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Cite This: J. Phys. Chem. Lett. 2019, 10, 1845−1851

Transient and Enduring Electronic Resonances Drive Coherent Long Distance Charge Transport in Molecular Wires Alessandro Landi,† Raffaele Borrelli,‡ Amedeo Capobianco,† and Andrea Peluso*,† †

Dipartimento di Chimica e Biologia, Università di Salerno, I-84084 Fisciano, Salerno, Italy Department of Agricultural, Forestry and Food Science, University of Torino, Via Leonardo da Vinci 44, I-10095 Grugliasco, Italy

‡

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S Supporting Information *

ABSTRACT: It is shown that the yields of oxidative damage observed in double-stranded DNA oligomers consisting of two guanines separated by adenine−thymine (A:T)n bridges of various lengths are reliably accounted for by a multistep mechanism, in which transient and nontransient electronic resonances induce charge transport and solvent relaxation stabilizes the hole transfer products. The proposed multistep mechanism leads to results in excellent agreement with the observed yield ratios for both the short and the long distance regime; the almost distance independence of yield ratios for longer bridges (n ≥ 3) is the consequence of the significant energy decrease of the electronic levels of the bridge, which, as the bridge length increases, become quasi-degenerate with those of the acceptor and donor groups (enduring resonance). These results provide significant guidelines for the design of novel DNA sequences to be employed in organic electronics.

L

molecules are in temporary resonance with each other. Such a flickering resonance could create a band-like state that promotes ballistic (i.e., at nearly constant velocity) motion of the hole through the bridge. Because by increasing the bridge length the probability of bringing all states in resonance drops exponentially, as more sites are involved, hole transfer at shorter distances could occur via the flickering resonance mechanism, which differs from tunneling inasmuch as bridge states can be occupied. Herein, following to some extent the above lines and also those recently explored by Parson,19−22 we model HT in the DNA oligomers studied by Giese,9 using a multistep kinetic model that consists of (i) an activation step that brings the donor and the acceptor groups into electronic degeneracy, (ii) the elementary electron transfer step between resonant donor and acceptor groups, and (iii) relaxation of all of the nonequilibrium species to their minimum-energy structures (including solvent) and formation of hole transfer products. The multistep mechanism is shown in Scheme 1, where dG+(T)nGGG and d-G(T)nGGG+ denote the initial state with the charge localized on the single G and the final HT state, respectively; [d-G+(T)nGGG]* and [d-G(T)nGGG+]* denote the ensembles of structures in which the hole donor and acceptor are in vibronic resonance with each other, and PG and PGGG denote the products of oxidative damage occurring at G and GGG. Giese’s work is particularly appealing for our purposes because it is one, among a few other papers,11 that provides the values of yield ratios of the water trapping products of hole transfer for a large number of oligomers,

ong-range charge transport in DNA has attracted much interest since its discovery1 because it is expected to play a crucial role in mutagenesis and carcinogenesis2,3 and because it enables the use of DNA for the development of environmentally friendly materials for organic electronics, hopefully leading to biocompatible and biodegradable devices. Indeed, DNA has found applications as dielectric material in field-effect transistors4−6 and more recently in organic light-emitting diodes.7,8 A large body of experimental evidence has shown that hole transport (HT) in DNA exhibits two regimes, one for shorter distances, which exponentially depends on the donor−acceptor distance, and the other for longer distances, which exhibits a much weaker distance dependence.9−12 That behavior is well represented by Giese’s experiment on double-stranded dG(T)nGGG oligomers,9 (T = thymine, G = guanine), in which a hole is injected onto the single G site via photoexcitation of a 4′-acylated nucleotide, and the yields of oxidation products formed at the initial site (PG) and at the trap site (PGGG) are revealed. For shorter sequences, up to n = 3, the product ratio PGGG/PG drops by a factor of ca. 8 for each additional adenine−thymine (A:T) step; in longer sequences, for n = 4− 7, the PGGG/PG ratio exhibits a much weaker distance dependence, whereas for n = 7−16, no change in PGGG/PG has been observed. These results have been interpreted in terms of a switch of HT mechanism from coherent superexchange to a thermally induced multistep or multirange hopping mechanism.10,13−17 A different and interesting point of view has been provided by Beratan and co-workers,18 who proposed that long-range charge transport could occur upon formation, via structural and energy fluctuations, of a transient state in which the electronic states of the redox groups and of the bridge © XXXX American Chemical Society

Received: March 7, 2019 Accepted: April 2, 2019 Published: April 2, 2019 1845

DOI: 10.1021/acs.jpclett.9b00650 J. Phys. Chem. Lett. 2019, 10, 1845−1851

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The Journal of Physical Chemistry Letters Scheme 1

introduced to take into account the oscillations of the electronic couplings HSm caused by interbase oscillations along the ith normal coordinate.48 The time-dependent wave function is written as

quantities which are of fundamental importance to test the reliability of the proposed multistep mechanism. For other systems, direct rate data have been provided, but in most of them, other effects arise that make theoretical simulations more involved.23,24 The DNA oligomers studied here contain several consecutive A:T steps that are known to provide structural rigidity to DNA tracts.25−27 We can thus assume that step 1 is mainly controlled by solvent motion rather than by backbone reorganization. The role played by the solvent in charge transport in DNA is still a controversial issue; solvent reorganization energies ranging from 0.5 to 3 eV have been reported in the literature.10,16,28−40 Our own computations, based on a nonequilibrium continuum polarizable model,41 yield a reorganization energy of the order of ca. 2 eV (see the Supporting Information (SI)), in line with recent estimates that also include explicit water molecules.42 The reverse of step 1 as well as that of step 3 consists of solvent response to a nonequilibrium charge distribution of the solute. Those relaxation processes have been deeply investigated by time-dependent spectroscopy. Pump−probe experiments have shown that in water solution solvent relaxation occurs on time scale of a few tens of femtoseconds;43 we will use those results to set the values of the rate constants. Step 2 is the elementary hole transfer step, which is instead mainly governed by the nuclear motion of nucleobases. Because of the local rigidity of the DNA backbone, we can make the reasonable assumption that the elements of the ensemble of the activated HT reactants differ from each other only for solvent configurations, which are irrelevant for calculation of the rates of the elementary HT step, so that dynamics computations can be carried out for only one of the typical equilibrium configurations of the different oligonucleotides. The model Hamiltonian describing charge transfer in Giese’s systems is defined following well-established approaches;17,38,44−47 we consider as many electronic states as the number of nucleobases that can be in the neutral or charged state and treat local nuclear motion in the harmonic approximation. The electronic states are a set of vibronic diabatic states |S , vS̅ ⟩, where S indicates the nucleobase on which the hole is fully localized and vS̅ denotes the multidimensional vector of the vibrational eigenstates of the S th diabatic state, including the vibrational states of all units, both those of the positively charged nucleobase and those of the other units in the neutral form. With the above notation, the Hamiltonian has the form /=

|Ψ(t )⟩ =

ij y jjH + ∂HSm Q zzz|S , v ⟩⟨m , v | + c.c. jj Sm z S̅ m̅ ∂Q i i z{ S, m , vS̅ , vm̅ k

(2)

and the expansion coefficients are determined by solving the time-dependent Schrödinger equation with the appropriate initial conditions. For that task, we used a robust quantum mechanical approach, significantly different from those used previously, inasmuch as we adopt (i) a multiset description of the wave function, which allows for use of the optimal vibrational basis set of each electronic state, at the cost of evaluating Franck−Condon (FC) factors;45,49 (ii) an effective strategy, borrowed from spectral band shape calculations,50−53 for generating reduced Hilbert spaces, in which the dynamical problem can be solved with numerically exact wave packet methodologies, see the SI for more details; (iii) a rigorous selection, via Duschinsky’s affine transformation, of the normal modes that are mostly coupled to hole transfer; and (iv) a new set of parameters inferred from experimental data and from ad hoc reliable computations. Adopted hole site energies and intrastrand electronic couplings for stacked nucleobases are reported in Table 1. Table 1. Hole Site Energies (EY, eV, Relative to the G/G+ Pair) and Electronic Coupling Parameters for Stacked Base Pairs (HYX, eV)a

a

Y

EY

HYG

HYA

HYC

HYT

G A C T

0.00 0.43 0.68 0.70

0.09 0.12 0.25 0.13

0.16 0.27 0.18 0.08

0.23 0.15 0.12 0.09

0.16 0.09 0.13 0.12

All values are from ref 54.

All values have been inferred from experimental data and have been further confirmed by DFT computations including both the sugar−phosphate ionic backbone and the effects of the aqueous environment.54 The reliability of the set of parameters of Table 1 has been extensively tested on several DNA oligomers, providing oxidation potentials in excellent agreement with the available experimental observations.55−59 The interstrand coupling element for G-A has been set to 0.012 eV.60 The nuclear modes mostly coupled to HT have been selected from Duschinsky transformation of the normal modes of each redox half-pair. Equilibrium position displacements of normal vibrations and intramolecular reorganization energies are reported in Table S1 of the SI. More than 80% of the intramolecular reorganization energy is recovered by considering 13 modes for the G/G+ half-pair, 6 for T/T+, and 8 for A/A+. We explicitly considered those nuclear degrees of

S

i

+

S

S, vS̅

∑ E v ̅ |S, vS̅ ⟩⟨S, vS̅ | S, vS̅

∑ CS, v ̅ (t )|S, vS̅ ⟩

∑

(1)

where E vS̅ is the energy of the vibronic state |S , vS̅ ⟩ and HSm is the electronic coupling term. The last term of eq 1 is 1846


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The Journal of Physical Chemistry Letters Table 2. Converged Transition Times (ps) for Intrastrand and Interstrand HT in G(T)nGGG Oligonucleotidesa intrastrand

interstrand

s

ms

T

TT

(T)3

(T)4

A

AA

(A)3

(A)4

(A)5

(A)6

(A)7

3

533

0.53

4.5

37.8

315

30

42

49

55

62

68

71b

a

s is the number of simultaneously excited modes, and ms denotes the maximum vibrational quantum number used for the sth subspace. bm3 = 2.

freedom in dynamics, with the vibrational frequencies and equilibrium position displacements reported in Table S1. As concerns the interbase oscillators that modulate electronic coupling elements, we performed a Duschinsky analysis for the single-stranded ss-5′-TTTATT-3′/5′-TTTA+TT-3′ half-pair and noted that most of displaced modes that involve interbase motion fall in a narrow frequency range centered at ca. 100 cm−1, with average displacements of about 0.1 Å amu1/2; those average values have been used in the present quantum dynamics computations, considering in dynamics only one mode per each nucleobase. Following previous works showing that the derivative of the electronic coupling is of the same order of magnitude as the electronic coupling itself,61−64 we set ∂/S, m/∂Q i = 0.1 eV/Å for any nucleobase pair. The results of HT dynamics in d-G(T)nGGG oligomers are summarized in Table 2, where computed transition times, taken at the complete population of the final state, i.e., when the hole is fully localized on the GGG site, are reported. For all of the dynamics simulations, the initial state is the vibronic ground state localized at the single G site because the vibrational frequencies of the nucleobase modes that are effectively coupled to hole motion exceed the thermal quantum at room temperature. The effect of thermal populations of the initial states on hole transfer rates could arise from the lowfrequency interbase modes that modulate the electronic coupling between nucleobases; we have neglected this effect, which can be treated in a numerically efficient way in the Hilbert space using approaches based on thermo field dynamics theory.65,66 Intrastrand transition times have been computed by considering that HT can take place only on the G(T)nGGG strand, without including interstrand electronic couplings. Computations predict that for shorter oligonucleotides, n = 1− 3, HT transition times increase by a factor of ∼8.4 for each additional A:T step; see Figure 1. The process that takes place is a genuine superexchange, inasmuch as the charge is never localized over the bridge. The logarithmic plot of the computed transition times against the distance between G and GGG nucleobases reported in Figure 1a yields a straight line with a slope of β = 0.63 Å−1 of the Marcus−Levich− Jortner equation,67 to be compared with β = 0.6 Å−1 obtained from experimental results.9 For n = 4, the predicted HT transition time along the single strand lines up with those for n = 1−3 in the logarithmic plot of transition times vs tunneling distance (see Figure 1), in contrast with the clear change in the slope observed experimentally. We have thus considered also the possibility that hole transfer is mediated by the adenine units of the complementary strand, as originally proposed by Giese.9 To reduce the computational load, the contribution of the excited vibrational states of the electronic states with the hole localized on the thymine units has been neglected in these simulations, leaving only electronic effects. Transition times for the interstrand HT are reported in Table 2. Interstrand HT is significantly slower than intrastrand

Figure 1. Top: Predicted rate constants k23 for the HT step in dG(T)nG as a function of the number of bases separating the donor and the acceptor G sites; green diamonds: intrastrand; violet circles: interstrand HT via the (A)n bridge of the complementary strand. Bottom: Predicted (black triangles) and experimental9 (red squares) yield ratios for the HT in d-G(T)nG as a function of the number of bases separating the donor and the acceptor G sites.

up to n = 2 and becomes about 1 order of magnitude faster for n = 4. Our computations predict that HT rates are almost distance-independent, in substantially good agreement with experimental results. The analysis of time-dependent populations reveals that even in these longer strands hole transfer occurs via superexchange, the electronic resonance of the acceptor and donor groups being the real driving force for hole motion. The different distance dependence of HT rates in the short and long distance regimes results from the balance of two effects: the decreasing energy barrier as the number of bridging adenines increases and the increasing tunneling distance. In Figure 2, the electronic hole site energies of the nucleobases as a function of the bridge units n have been reported; the electronic site energies of the A bridge significantly decrease as n increases because of the stacking interactions among resonant states. For n ≥ 4, the lowest electronic energy of the bridge nucleobases becomes almost degenerate with those of the resonant acceptor and donor groups, enabling a charge transport almost independent of the distance. Those results are consistent not only with Giese’s experiment but also with other pieces of experimental evidence: (i) Barton and co-workers have reported kinetic data of charge transport in DNA that suggest coherent charge transport over six or more adenines and suggested that such a coherent transport regime could grow in significance as the distance between the donor and acceptor increases;68−70 (ii) voltammetric measurements of Gfree oligomers clearly show a progressive lowering of the 1847


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underestimated, 195 vs ≥250; that could be due to the assumption that k12 is independent of the length of the bridge, which is expected to hold only when the bridge reaches a certain length. Finally, we remark that in the present treatment thermal effects on the rates of the hole transfer step between resonant states have been neglected; it is well possible that for long bridges, for which enduring electronic resonances between the donor and the bridge states are established, the bridge states could be thermally populated and the HT mechanism could switch from superexchange to resonant transport, in line with previous works.16,71,75 The results reported here are of great importance for use of DNA in organic electronics, inasmuch as they provide guidelines for properly choosing a DNA sequence to employ in devices. Indeed, we have shown that enduring electronic resonances can establish between states that carry a hole; oligomers in which multiple A units are properly alternated by G units, possibly on the same strand, could give rise to sequences that are expected to transfer a hole over a long distances in a fast and efficient way. We believe that the computational approach and the set of parameters used here are broadly applicable to other DNA oligomers whose HT properties have been very well characterized in the past;11,23,24 work is in progress along that line.

Figure 2. Electronic hole site energies of the nucleobases for different bridge lengths. Blue line = A; dashed red line = T; black dashed thick line = initial G; magenta dotted thick line = final three G’s.

potential of the first anodic peak as the number of consecutive adenines increases;58 (iii) it is well-known from X-ray analysis that consecutive A units confer particular rigidity to DNA tracts, likely because of larger stacking interactions.25,26 The importance of delocalized bridge states in long-range charge transport emerges also in other theoretical treatments, such as the recently proposed quantum unfurling mechanism, in which the localized hole at the single G site is transferred by thermal fluctuations into a delocalized state, spread over the entire molecular bridge. Because in the quantum unfurling mechanism the propagation through the bridge occurs in a single step, the transfer rate becomes independent of the bridge length.71 The mechanism proposed here differs from quantum unfurling, inasmuch as charge transport is not triggered by thermal fluctuations but rather by the quasielectronic degeneracy between a low-energy-lying delocalized bridge state and the electronic states of the donor and acceptor sites, a consequence of the high electronic coupling among stacked adenine nucleobases. For a deeper comparison with the experimental results, the set of ordinary differential equations (ODEs) of the kinetic model of Scheme 1 have been numerically solved to obtain the yield ratios PG/PGGG. The rates of the elementary HT steps k23 have been taken from transition times when initial populations drop to 1/e; because for that step reactants and products are at the same energy, we set k23 = k32. As discussed above, we set k21 = krel = 1013 s−1 from time-dependent measurements of the solvent relaxation rates.43,72 Finally, we set kP = 107 s−1, taken from the rate of deprotonation of Guo•+, likely the first step of formation of the products of oxidative damage, which have been reliably measured in double-stranded DNA.73,74 There is no experimental information about k12, which we thus keep as an adjustable parameter to be inferred from experimental results. Numerical resolutions of the set of ODEs show that the experimental yield ratios are compatible with only k12 on the order of 1010 s−1; setting k12 = 1 × 1010 s−1 and assuming that the rate of the activation process is independent of the bridge length, computations yield PGGG/PG ratios of 35, 4.4, and 3 for n = 2−4, respectively, to be compared with the observed values: 30 ± 5, 4.0 ± 0.5, and 3.5 ± 0.5. For n = 4−7, the predicted yield ratios show a very weak dependence on n, in excellent agreement with the experimental results; see Figure 1b. For n = 1, the computed yield ratio is slightly

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COMPUTATIONAL METHODS Franck−Condon integrals have been computed by using a development version of the MolFC package.76 Full details about implementation of the generating function approach can be found in ref 77. In all FC calculations, the curvilinear coordinate representation of the normal modes has been adopted to prevent that a large displacement of an angular coordinate could reflect into large shifts of the involved bond distances. That is unavoidable in rectilinear Cartesian coordinates and requires the use of high-order anharmonic potentials for its correction.76,78 The numerical solution of the time-dependent Schrödinger equation has been carried out with an orthogonalized Krylov subspace method.79,80 To further reduce the overall computational costs, computation of the FC integrals has been carried out using the separate-mode approximation, which allows factorization of the multidimensional FC integrals into the product of onedimensional integrals. That approximate method for fast FC computations corresponds roughly to neglecting the offdiagonal terms of the Duschinsky transformation but taking into account the changes of the vibrational frequencies of the vibrational modes.65,81 A convergence threshold of ca. 5% on transition times has been used in all dynamics simulations; that level of accuracy requires including in computations subspaces up to three vibrations simultaneously excited; further details about convergence in dynamics are reported in the SI. The set of coupled ODEs has been solved by using the Dormand−Prince method of order 4,82 a member of the Runge−Kutta family of ODE solvers. Equilibrium geometries, normal coordinates, and vibrational frequencies of nucleobases and ss-5′-TTTATT-3′ (A1) oligonucleotide in their neutral and cationic forms were computed at the density functional theory (DFT) level by the B3LYP functional.83,84 For A1, the correction for dispersion forces proposed by Grimme was included in DFT computations.85 The unrestricted formalism was used for 1848


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(8) Gomez, E. F.; Venkatraman, V.; Grote, J. G.; Steckl, A. J. Exploring the Potential of Nucleic Acid Bases in Organic Light Emitting Diodes. Adv. Mater. 2015, 27, 7552−7562. (9) Giese, B.; Amaudrut, J.; Köhler, A.-K.; Spormann, M.; Wessely, S. Direct Observation of Hole Transfer through DNA by Hopping between Adenine Bases and by Tunneling. Nature 2001, 412, 318− 320. (10) Bixon, M.; Giese, B.; Wessely, S.; Langenbacher, T.; MichelBeyerle, M. E.; Jortner, J. Long-Range Charge Hopping in DNA. Proc. Natl. Acad. Sci. U. S. A. 1999, 96, 11713−11716. (11) Joseph, J.; Schuster, G. B. Emergent Functionality of Nucleobase Radical Cations in Duplex DNA: Prediction of Reactivity Using Qualitative Potential Energy Landscapes. J. Am. Chem. Soc. 2006, 128, 6070−6074. (12) Kanvah, S.; Joseph, J.; Schuster, G. B.; Barnett, R. N.; Cleveland, C. L.; Landman, U. Oxidation of DNA: Damage to Nucleobases. Acc. Chem. Res. 2010, 43, 280−287. (13) Renger, T.; Marcus, R. A. Variable Range Hopping Electron Transfer Through Disordered Bridge States: Application to DNA. J. Phys. Chem. A 2003, 107, 8404−8419. (14) Bixon, M.; Jortner, J. Incoherent Charge Hopping and Conduction in DNA and Long Molecular Chains. Chem. Phys. 2005, 319, 273−282. (15) Grozema, F. C.; Tonzani, S.; Berlin, Y. A.; Schatz, G. C.; Siebbeles, L. D. A.; Ratner, M. A. Effect of Structural Dynamics on Charge Transfer in DNA Hairpins. J. Am. Chem. Soc. 2008, 130, 5157−5166. (16) Renaud, N.; Berlin, Y. A.; Lewis, F. D.; Ratner, M. A. Between Superexchange and Hopping: An Intermediate Charge-Transfer Mechanism in polyA-polyT DNA Hairpins. J. Am. Chem. Soc. 2013, 135, 3953−3963. (17) Levine, A. D.; Iv, M.; Peskin, U. Length-Independent Transport Rates in Biomolecules by Quantum Mechanical Unfurling. Chem. Sci. 2016, 7, 1535−1542. (18) Zhang, Y.; Liu, C.; Balaeff, A.; Skourtis, S. S.; Beratan, D. N. Biological Charge Transfer via Flickering Resonance. Proc. Natl. Acad. Sci. U. S. A. 2014, 111, 10049−10054. (19) Parson, W. W. Vibrational Relaxations and Dephasing in Electron-Transfer Reactions. J. Phys. Chem. B 2016, 120, 11412− 11418. (20) Parson, W. W. Effects of Free Energy and Solvent on Rates of Intramolecular Electron Transfer in Organic Radical Anions. J. Phys. Chem. A 2017, 121, 7297−7306. (21) Parson, W. W. Electron-Transfer Dynamics in a Zn-PorphyrinQuinone Cyclophane: Effects of Solvent, Vibrational Relaxations, and Conical Intersections. J. Phys. Chem. B 2018, 122, 3854−3863. (22) Parson, W. W. Temperature Dependence of the Rate of Intramolecular Electron Transfer. J. Phys. Chem. B 2018, 122, 8824− 8833. (23) Lewis, F. D.; Liu, X.; Liu, J.; Miller, S. E.; Hayes, R. T.; Wasielewski, M. R. Direct Measurement of Hole Transport Dynamics in DNA. Nature 2000, 406, 51−53. (24) Kawai, K.; Majima, T. Hole Transfer Kinetics of DNA. Acc. Chem. Res. 2013, 46, 2616−2625. (25) El Hassan, M. A.; Calladine, C. R. Conformational Characteristics of DNA: Empirical Classifications and a Hypothesis for the Conformational Behaviour of Dinucleotide Steps. Philos. Trans. R. Soc. A 1997, 355, 43−100. (26) Calladine, C. R.; Drew, H. R.; Luisi, B. F.; Travers, A. A. Understanding DNA, 3rd ed.; Elsevier Academic Press: Oxford, 2004; Chapter 3. (27) Capobianco, A.; Peluso, A. The Oxidization Potential of AA Steps in Single Strand DNA Oligomers. RSC Adv. 2014, 4, 47887− 47893. (28) Harriman, A. Electron Tunneling in DNA. Angew. Chem., Int. Ed. 1999, 38, 945−949. (29) Lewis, F. D.; Kalgutkar, R. S.; Wu, Y.; Liu, X.; Liu, J.; Hayes, R. T.; Miller, S. E.; Wasielewski, M. R. Driving Force Dependence of

charged species. Solvent (water) effects were included in all computations and were estimated by the polarizable continuum model (PCM).86 The 6-311++G(d,p) basis set was employed throughout, except for A1, treated with the SV(P) basis set; see also ref 87 and the SI. The Gaussian package was used for DFT computations.88

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b00650. Detailed description of the model Hamiltonian; method selection of active normal modes and equilibrium position displacements (Table S1); transition times for different Hilbert subspaces; and reorganization energies of all of the nucleobases studied (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Alessandro Landi: 0000-0003-3627-5535 Raffaele Borrelli: 0000-0002-0060-4520 Amedeo Capobianco: 0000-0002-5157-9644 Andrea Peluso: 0000-0002-6140-9825 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are indebted to Prof. W. H. Parson for careful reading of the manuscript and for helpful discussions. We acknowledge the HP10CK2XJU and HP10CYW18T CINECA awards under the ISCRA initiative for the availability of highperformance computing resources and support and the European Community, COST action CM1201. Financial support of Ministero dell’Istruzione, dell’Università e della Ricerca, Università di Salerno, and Università di Torino is gratefully acknowledged.

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REFERENCES

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